ACTA issues

Isometries on certain non-complete vector-valued function spaces

Mojtaba Mojahedi, Fereshteh Sady

Acta Sci. Math. (Szeged) 85:3-4(2019), 613-627

Abstract. Surjective, not necessarily linear isometries $T\colon {\rm AC}(X,E) \to {\rm AC}(Y,F)$ between vector-valued absolutely continuous functions on compact subsets $X$ and $Y$ of the real line have recently been described as generalized weighted composition operators. The target spaces $E$ and $F$ are strictly convex normed spaces. In this paper, we assume that $X$ and $Y$ are compact Hausdorff spaces and $E$ and $F$ are normed spaces, which are not assumed to be strictly convex. We describe (with a short proof) surjective isometries $T\colon (A,\|\cdot \|_A) \to (B,\|\cdot \|_B)$ between certain normed subspaces $A$ and $B$ of $C(X,E)$ and $C(Y,F)$, respectively. We consider three cases for $F$ with some mild conditions. The first case, in particular, provides a short proof for the above result, without assuming that the target spaces are strictly convex. The other cases give some generalizations in this topic. As a consequence, the results can be applied, for isometries (not necessarily linear) between spaces of absolutely continuous vector-valued functions, (little) Lipschitz functions and also continuously differentiable functions.

DOI: 10.14232/actasm-018-092-6

AMS Subject Classification (1991): 47B38, 47B33; 46J10

Keyword(s): real-linear isometries, vector-valued function spaces, T-sets

Received October 29, 2018 and in final form March 12, 2019. (Registered under 92/2018.)